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arXiv · 0901.1842

Small gain theorems for large scale systems and construction of ISS Lyapunov functions

Sergey N. Dashkovskiy·Björn S. Rüffer·Fabian R. Wirth
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Abstract

We consider interconnections of n nonlinear subsystems in the input-to-state stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems. The results are obtained in a general formulation of ISS, the cases of summation, maximization and separation with respect to external gains are obtained as corollaries.

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Sergey N. Dashkovskiy, Björn S. Rüffer, Fabian R. Wirth. 2009-01-13. Small gain theorems for large scale systems and construction of ISS Lyapunov functions. https://doi.org/10.1137/090746483

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